Chapter 10.4 - Number of Moles of Atoms and Molecules

In the previous section, we saw GMM. In this section we will see the details about 'mole':

We have seen that,
• 1 GAM of any element will contain N_A atoms
    ♦ That is., 6.022×10²³ atoms
• 1 GMM of any element/compound will contain N_A molecules
    ♦ That is., 6.022×10²³ molecules
■ This 6.022×10²³ is a very important number in chemistry. So it is given a special name: mole.
Let us see a comparison:
• Consider a group of apples. If there are 12 apples in that group, we say: A dozen apples
• Consider a group of atoms. If there are 6.022×10²³ atoms in that group, we say: A mole atoms
• Consider a group of molecules. If there are 6.022×10²³ molecules in that group, we say: A mole molecules
• Consider a group of ions. If there are 6.022×10²³ ions in that group, we say: A mole ions
■ While using mole, it is important to specify the name of the particle. That is., whether it is atom, molecule or ion.

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Based on the above discussion, a relation can be established between two sides. One side consists of the mole. The other side consists of Avogadro number (N_A), GAM and GMM. The fig.10.6 below illustrates the relation:

Fig.10.6

■ The definition for mole can be written as follows:
The amount of any substance containing 6.022×10²³ particles is called one mole (1 mole or 1 mol)

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The concept will become more clear when we see some solved examples 

Solved example 10.3
A sample of water contains 10,000 water molecules. How many moles of molecules are present in it?
Solution:
1. One mole of molecules will contain 6.022×10²³ molecules.
2. So no. of moles in 10000 molecules = ¹⁰⁰⁰⁰⁄_{(6.022 × 10²³)} 

Solved example 10.4
Find the number of moles in the following:
(i) 1,00,000 CO₂ molecules
(ii) 1,00,000 H₂ molecules
(iii) 6.022×10²³ glucose molecules   
Solution (i):
1. One mole of molecules will contain 6.022×10²³ molecules.
2. So no. of moles in 1,00,00 molecules = ¹⁰⁰⁰⁰⁰⁄_{(6.022 × 10²³)} 
That is., 1,00,000 CO₂ molecules will contain [¹⁰⁰⁰⁰⁰⁄_{(6.022 × 10²³)}] moles
Solution (ii):
1. One mole of molecules will contain 6.022×10²³ molecules.
2. So no. of moles in 1,00,00 molecules = ¹⁰⁰⁰⁰⁰⁄_{(6.022 × 10²³)} .
That is., 1,00,000 H₂ molecules will contain [¹⁰⁰⁰⁰⁰⁄_{(6.022 × 10²³)}] moles
Solution (iii):
1. One mole of molecules will contain 6.022×10²³ molecules.
2. So no. of moles in 6.022×10²³ molecules = ^{(6.022 × 1023)}⁄_{(6.022 × 10²³)} = 1
That is., 6.022×10²³ glucose molecules will contain 1 mole

Solved example 10.5
Find the number of moles of atoms in 160 grams of oxygen
Solution:
1. One mole of atoms will contain 6.022×10²³ atoms.
• So no. of moles of atoms = ^{(Total no. of atoms)}⁄_{(6.022 × 10²³)} .
2. But we are not given 'Total no. of atoms'. We have to calculate it from the given 'total mass'
• Given total mass = 160 g
3. 1 GAM of oxygen will contain 6.022×10²³ atoms of oxygen
• 1 GAM of oxygen = 16 g
• No. of GAMs in 160 g = ¹⁶⁰⁄₁₆ = 10
4. So no. of atoms in 160 g = 10 × 6.022×10²³
5. Substituting this in (1) we get:
No. of moles of atoms = ^{(Total no. of atoms)}⁄_{(6.022 × 10²³)} = ^{(10×6.022×1023)}⁄_{(6.022×10²³)} = 10
• We can write the above result in the form of an equation:
Eq.10.1:
No. of moles of atoms/molecules = ^{(Total no. of atoms/molecules)}⁄_NA

■ Another method using basic information:
1. We have:
No. of atoms × Mass (in grams) of one atom = Given total Mass (in grams)
⇒ No. of atoms = ^{Given total Mass (in grams)}⁄_{Mass (in grams) of one atom}.  
2. Given total mass = 160 grams
3. Mass (in u) of one atom of oxygen = 16 u
4. But 1 u = 1.6605×10^-24 grams  (Details here)
So 16 u = 16 × 1.6605×10^-24 grams
5. Substituting this value in (1) we get:
No. of atoms = ^{Given total Mass (in grams)}⁄_{Mass (in grams) of one atom} 
= ¹⁶⁰⁄_{(16×1.6605×10^-24)} = ¹⁰⁄_{(1.6605×10^-24)}
6. No. of moles of atoms = ^{No. of atoms}⁄_{(6.022×10²³)} 
= [¹⁰⁄_{(1.6605×10^-24)}] ÷ 6.022×10²³ = [¹⁰⁄_{(1.6605×10^-24)}] × [¹⁄_{(6.022 × 10²³)}] 
7. But ¹⁄_{(6.022 × 10²³)} = 1.6605×10^-24. So numerator and denominator will cancel each other.  We get 10 as the answer

Solved example 10.6
Find the number of moles of molecules in 160 grams of oxygen
Solution:
1. One mole of molecules will contain 6.022×10²³ molecules.
• So no. of moles = ^{(Total no. of molecules)}⁄_{(6.022 × 10²³)} .
2. But we are not given 'Total no. of molecules'. We have to calculate it from the given 'total mass'
• Given total mass = 160 g
3. 1 GMM of oxygen will contain 6.022×10²³ molecules of oxygen
• 1 GMM of oxygen = 32 g
• No. of GMMs in 160 g = ¹⁶⁰⁄₃₂ = 5
4. So no. of molecules in 160 g = 5 × 6.022×10²³
5. Substituting this in (1) we get:
No. of moles = ^{(Total no. of molecules)}⁄_{(6.022 × 10²³)} = ^{(5×6.022×1023)}⁄_{(6.022×10²³)} = 5

■ Another method using basic information:
1. We have:
No. of molecules × Mass (in grams) of one molecule = Given total Mass (in grams)
⇒ No. of molecules = ^{Given total Mass (in grams)}⁄_{Mass (in grams) of one molecule} .  
2. Given total mass = 160 grams
3. Mass (in u) of one molecule of oxygen = 32 u
4. But 1 u = 1.6605×10^-24 grams  (Details here)
So 32 u = 32 × 1.6605×10^-24 grams
5. Substituting this value in (1) we get:
No. of molecules = ^{Given total Mass (in grams)}⁄_{Mass (in grams) of one atom} 
= ¹⁶⁰⁄_{(32×1.6605×10^-24)} = ⁵⁄_{(1.6605×10^-24)}
6. No. of moles of molecules = ^{No. of molecules}⁄_{(6.022×10²³)} 
= [⁵⁄_{(1.6605×10^-24)}] ÷ 6.022×10²³ = [⁵⁄_{(1.6605×10^-24)}] × [¹⁄_{(6.022 × 10²³)}] 

7. But ¹⁄_{(6.022 × 10²³)} = 1.6605×10^-24. So numerator and denominator will cancel each other.  We get 5 as the answer

Solved example 10.7
Find the number of moles of molecules in 220 grams of CO₂
Solution:
1. One mole of molecules will contain 6.022×10²³ molecules.
• So no. of moles = ^{(Total no. of molecules)}⁄_{(6.022 × 10²³)}
2. But we are not given 'Total no. of molecules'. We have to calculate it from the given 'total mass'
• Given total mass = 220 g
3. 1 GMM of CO₂ will contain 6.022×10²³ molecules of CO₂
• 1 GMM of CO₂ = (1×12) + (2×16) = 12 + 32 = 44 g
• No. of GMMs in 220 g = ²²⁰⁄₄₄ = 5
4. So no. of molecules in 220 g = 5 × 6.022×10²³
5. Substituting this in (1) we get:
No. of moles = ^{(Total no. of molecules)}⁄_{(6.022 × 10²³)} = ^{(5×6.022×1023)}⁄_{(6.022×10²³)} = 5

■ Another method using basic information:
1. We have:
No. of molecules × Mass (in grams) of one molecule = Given total Mass (in grams)
⇒ No. of molecules = ^{Given total Mass (in grams)}⁄_{Mass (in grams) of one molecule} .  
2. Given total mass = 220 grams
3. Mass (in u) of one molecule of CO₂ = (1×12) + (2×16) = 12 + 32 = 44 u
4. But 1 u = 1.6605×10^-24 grams  (Details here)
So 44 u = 44 × 1.6605×10^-24 grams
5. Substituting this value in (1) we get:
No. of molecules = ^{Given total Mass (in grams)}⁄_{Mass (in grams) of one atom} 
= ²²⁰⁄_{(44×1.6605×10^-24)} = ⁵⁄_{(1.6605×10^-24)}
6. No. of moles of molecules = ^{No. of molecules}⁄_{(6.022×10²³)} 
= [⁵⁄_{(1.6605×10^-24)}] ÷ 6.022×10²³ = [⁵⁄_{(1.6605×10^-24)}] × [¹⁄_{(6.022 × 10²³)}] 

7. But ¹⁄_{(6.022 × 10²³)} = 1.6605×10^-24. So numerator and denominator will cancel each other.  We get 5 as the answer

Solved example 10.8
Find 
(i) The number of moles of atoms in 700 grams of nitrogen
(ii) The number of moles of molecules in 700 grams of nitrogen
(iii) Number of moles of sugar molecules in 1 kg of sugar (C₁₂H₂₂O₁₁) and the number of moles of carbon atoms present in that 1 kg
(iv) Number of moles in 3.011×10²³ carbon atoms
Solution (i):
1. One mole of atoms will contain 6.022×10²³ atoms.
• So no. of moles = ^{(Total no. of atoms)}⁄_{(6.022 × 10²³)} .
2. But we are not given 'Total no. of atoms'. We have to calculate it from the given 'total mass'
• Given total mass = 700 g
3. 1 GAM of nitrogen will contain 6.022×10²³ atoms of nitrogen
• 1 GAM of nitrogen = 14 g
• No. of GAMs in 700 g = ⁷⁰⁰⁄₁₄ = 50
4. So no. of atoms in 700 g = 50 × 6.022×10²³
5. Substituting this in (1) we get:
No. of moles = ^{(Total no. of atoms)}⁄_{(6.022 × 10²³)} = ^{(50×6.022×1023)}⁄_{(6.022×10²³)} = 50

Solution (ii):
1. One mole of molecules will contain 6.022×10²³ molecules.
• So no. of moles = ^{(Total no. of molecules)}⁄_{(6.022 × 10²³)}
2. But we are not given 'Total no. of molecules'. We have to calculate it from the given 'total mass'
• Given total mass = 700 g
3. 1 GMM of nitrogen will contain 6.022×10²³ molecules of nitrogen
• 1 GMM of nitrogen = 28 g
• No. of GMMs in 700 g = ⁷⁰⁰⁄₂₈ = 25
4. So no. of molecules in 700 g = 25 × 6.022×10²³
5. Substituting this in (1) we get:
No. of moles = ^{(Total no. of molecules)}⁄_{(6.022 × 10²³)} = ^{(25×6.022×1023)}⁄_{(6.022×10²³)} = 25
Solution (iii):

1. One mole of molecules will contain 6.022×10²³ molecules.
• So no. of moles = ^{(Total no. of molecules)}⁄_{(6.022 × 10²³)}
2. But we are not given 'Total no. of molecules'. We have to calculate it from the given 'total mass'
• Given total mass = 1 kg = 1000 g
3. 1 GMM of C₁₂H₂₂O₁₁ will contain 6.022×10²³ molecules of C₁₂H₂₂O₁₁
• 1 GMM of C₁₂H₂₂O₁₁ = (12×12) + (22×1) + (11×16) = 144 + 22 + 176 = 342 g
• No. of GMMs in 1000 g = ¹⁰⁰⁰⁄₃₄₂ = 2.92
4. So no. of molecules in 1000 g = 2.92 × 6.022×10²³
5. Substituting this in (1) we get:
No. of moles = ^{(Total no. of molecules)}⁄_{(6.022 × 10²³)} = ^{(2.92×6.022×1023)}⁄_{(6.022×10²³)} = 2.92
6. From (4) we get the total number of sugar molecules as 2.92 × 6.022×10²³.

7. Each of these molecules will contain 12 carbon atoms. 
So total number of carbon atoms in the 1000 grams of sugar = 12 × 2.92 × 6.022×10²³.
8. So no. of moles of carbon atoms = ^{(Total no. of atoms)}⁄_{(6.022 × 10²³)} 
= ^{(12×2.92×6.022×1023)}⁄_{(6.022×10²³)} = 12×2.92 = 35.04 moles
Solution (iv):
1. One mole of atoms will contain 6.022×10²³ atoms.
• So no. of moles = ^{(Total no. of atoms)}⁄_{(6.022 × 10²³)}
2. We are given 'Total no. of atoms'. It is 3.011×10²³.
3. So no. of moles of carbon atoms = ^{(Total no. of atoms)}⁄_{(6.022 × 10²³)} 
= ^{(3.011×1023)}⁄_{(6.022×10²³)} = 0.5 moles

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So we now know how to convert mass into 'number of moles'. We can also convert a given number of atoms/molecules into 'number of moles'. In the next section we will see the reverse. That is., conversion of 'number of moles' into mass. 

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